Determine the infrared orbifold action in the level-zero Landau–Ginzburg theory

Determine the action of the orbifold group generator on the NS–NS and R–R ground states of the infrared limit of the level-zero Landau–Ginzburg theory with one massive chiral multiplet and superpotential W = Φ².

Background

The paper extrapolates the N=2 minimal model construction to level k=0 and compares it with the infrared limit of a Landau–Ginzburg theory whose superpotential W = Φ² is a mass term. In both the NS–NS and R–R sectors, the infrared theory has a single ground state, matching the proposed level-zero minimal-model representation.

Although the orbifold generator’s action on excited states follows from its action on the fields, the action on the two ground states is not fixed by that argument. Determining these phases would clarify how the level-zero Landau–Ginzburg description realizes the Arf-invariant or discrete-torsion freedom used in the Gepner construction.

References

The present authors are not aware of a logic in the language of the Landau--Ginzburg theory, however, to determine the aciton of the orbifold group generator on the two ground states; the orbifold genereator multiplies $(-1){p_0(\vec{x})}$ on the fields $\phi$, $\psi$ and $\tilde{\psi}$, so a state above a one-particle excitation must have an extra $(-1){p_0(\vec{x})}$ phase relatively to the state before the excitation, but this argument does not determine the response of the NS--NS and R--R ground states under the generator.

— Notes on Gepner Construction  (2608.26017 - Nishikawa et al., 26 Aug 2026) in Section 3.2, Reinterpretation of the factor (-1)^{Arf[s+p_0]}(-1)^{Arf[s]}